Wilf-equivalence for singleton classes

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Abstract

Write p1 p2 ... pm for the permutation matrix (δpi, j)m × m. Let Sn (M) be the set of n × n permutation matrices which do not contain the m × m permutation matrix M as a submatrix. In [R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin. 6 (1985) 383-406] Simion and Schmidt show bijectively that | Sn (123) | = | Sn (213) |. In the present work, we give a bijection from Sn (12 ... t pt + 1 ... pm) to Sn (t ... 21 pt + 1 ... pm). This result was established for t = 2 in [J. West, Permutations with forbidden subsequences and stack-sortable permutations, PhD thesis, MIT, Cambridge, MA, 1990] and for t = 3 in [E. Babson, J. West, The permutations 123 p4 ... pt and 321 p4 ... pt are Wilf-equivalent, Graphs Combin. 16 (2001) 373-380]. Moreover, if we think of n × n permutation matrices as transversals of the n by n square diagram, then we generalise this result to transversals of Young diagrams. © 2006 Elsevier Inc. All rights reserved.

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Backelin, J., West, J., & Xin, G. (2007). Wilf-equivalence for singleton classes. Advances in Applied Mathematics, 38(2), 133–148. https://doi.org/10.1016/j.aam.2004.11.006

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